Group

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Group
/ w, g7 T5 E- o- Y8 l* D6 K9 a( |A group is defined as a finite or infinite set of Operands
(called ``elements'') , , , ... that may be combined or ``multiplied'' via a Binary Operator
to form well-defined products and which furthermore satisfy the following conditions:
, Z: _6 }9 g0 j9 ?6 s1. Closure: If and are two elements in , then the product is also in .
2. Associativity: The defined multiplication is associative, i.e., for all , .
3. Identity: There is an Identity Element
! C( ~; Q! A2 n# Y/ i (a.k.a. , , or ) such that for every element .
3 `# S2 n' L0 U" q# c. ]4. Inverse: There must be an inverse or reciprocal of each element. Therefore, the set must contain an element such that for each element of .
A group is therefore a Monoid
for which every element is invertible. A group must contain at least one element.

* `" u4 k( V  B; [* @9 pThe study of groups is known as Group Theory
8 B  j5 G" h! [( T$ B4 l. If there are a finite number of elements, the group is called a Finite Group
and the number of elements is called the Order
of the group.
6 o3 Y- a' ]6 c# c+ z
$ d& f+ S$ d, ISince each element , , , ..., , and is a member of the group, group property 1 requires that the product
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(1)

" u/ k/ j8 f/ h. emust also be a member. Now apply to ,

(2) 8 _8 @) H" h+ H& Y

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3 M4 \, f! F  R! G# K3 T/ IBut
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2 w5 L4 d/ A3 p4 p# w4 n	(3)

so

(4)

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(5)

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" @& m4 x, N; _9 e/ A8 Dand

(6)

6 O7 i' T6 {$ x2 B" }

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